Could be factory security/vats or maybe even bad BCM
I've owned 2 '06 GTO's and my first one had that problem. It ended up being the key fob not communicating with the BCM. The security/vats system was keeping it from starting or would turn it off right after starting and it's very frustrating. I did some search on the forum and a lot of ppl who had similar problems said it was a bad BCM. A sure fire way to tell if your BCM is going bad is if your door locks don't work or only work once in a while. Why? I'm not sure, but that's what that means. My way around it was to go to shop that has an HP tuner and have them disarm the factory security/vats system and you won't have that problem anymore. It will start every time. Door locks probably won't work still, but at least now you'll have a running car! Another down side is that your keyless entry might not work either, but an after market alarm will cure that plus give you added security. I've done this to both GTO's and it ran me $20 at a shop. Takes literally 5 minutes for them to do or if you have one, you can do it yourself. If that doesn't work for some reason then I have no clue at all...